Connectivity and Hamiltonicity of Canonical Colouring Graphs of Bipartite and Complete Multipartite Graphs

Ruth Haas,Gary Macgillivray

doi:10.3390/a11040040

Ruth Haas, Gary Macgillivray

Open Access

https://doi.org/10.3390/a11040040

Copy DOI

Abstract

A k-colouring of a graph G with colours 1 , 2 , … , k is canonical with respect to an ordering π = v 1 , v 2 , … , v n of the vertices of G if adjacent vertices are assigned different colours and, for 1 ≤ c ≤ k , whenever colour c is assigned to a vertex v i , each colour less than c has been assigned to a vertex that precedes v i in π . The canonical k-colouring graph of G with respect to π is the graph Can k π ( G ) with vertex set equal to the set of canonical k-colourings of G with respect to π , with two of these being adjacent if and only if they differ in the colour assigned to exactly one vertex. Connectivity and Hamiltonicity of canonical colouring graphs of bipartite and complete multipartite graphs is studied. It is shown that for complete multipartite graphs, and bipartite graphs there exists a vertex ordering π such that Can k π ( G ) is connected for large enough values of k. It is proved that a canonical colouring graph of a complete multipartite graph usually does not have a Hamilton cycle, and that there exists a vertex ordering π such that Can k π ( K m , n ) has a Hamilton path for all k ≥ 3 . The paper concludes with a detailed consideration of Can k π ( K 2 , 2 , … , 2 ) . For each k ≥ χ and all vertex orderings π , it is proved that Can k π ( K 2 , 2 , … , 2 ) is either disconnected or isomorphic to a particular tree.

Highlights

One definition of a k-colouring of a graph G is as a function f : V ( G ) → {1, 2, . . . , k } such that f ( x ) 6= f (y) whenever xy ∈ E( G )
For an ordering π of the vertices of a graph G, the canonical k-colouring graph of G, denoted Canπ k (G), has vertex set equal to the set of canonical k-colourings of G with respect to π, with two of these being adjacent when they differ in the colour assigned to exactly one vertex
Since the complete multipartite graph Kn1,n2,...,nr is the join of K n1, K n2, . . . , K nr, our results show that there are vertex orderings π for which Canπ k ( Kn1,n2,...,nr ) is connected whenever k ≥ r

Summary

Introduction

One definition of a k-colouring of a graph G is as a function f : V ( G ) → {1, 2, . . . , k } such that f ( x ) 6= f (y) whenever xy ∈ E( G ). For an ordering π of the vertices of a graph G, the canonical k-colouring graph of G, denoted Canπ k (G), has vertex set equal to the set of canonical k-colourings of G with respect to π, with two of these being adjacent when they differ in the colour assigned to exactly one vertex. For every tree T there exists an ordering π of the vertices such that the canonical k-colouring graph of T with respect to π is Hamiltonian for all k ≥ 3. The canonical 3-colouring graph of the cycle Cn is disconnected for all vertex orderings π, while for each k ≥ 4 there exists an ordering π for which Canπ k (Cn ) is connected.

Unions and Joins

Bipartite Graphs

Conclusions